Transcript Slide 1

c[n]
Modern Sampling Methods
049033
Instructor: Yonina Eldar
Teaching Assistant: Tomer Michaeli
Spring 2009
c[n]
Sampling: “Analog Girl in a Digital World…”
Judy Gorman 99
Analog world
Digital world
Sampling
A2D
Signal processing
Denoising
Image analysis …
Reconstruction
D2A
(Interpolation)
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Applications
Sampling Rate Conversion
Common audio standards:
8 KHz (VOIP, wireless microphone, …)
11.025 KHz (MPEG audio, …)
16 KHz (VOIP, …)
22.05 KHz (MPEG audio, …)
32 KHz (miniDV, DVCAM, DAT, NICAM, …)
44.1 KHz (CD, MP3, …)
48 KHz (DVD, DAT, …)
…
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Applications
Image Transformations
Lens distortion correction
Image scaling
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Applications
CT Scans
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Applications
Spatial Superresolution
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Applications
Temporal Superresolution
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Applications
Temporal Superresolution
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Applications
Low-Rate Wide Band Conversion
5th order Chebyshev Type-I Filter
Spectrum Analyzer
Sign wavefor generator @ 54 MHz
M=32
Scope
Signal generator
carrier @ 500 MHz
(11 dBm)
Power splitter
(m=2 channels)
Mixer
Lowpass
filter
MATLAB ™
(reconstruction)
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Our Point-Of-View
The field of sampling was traditionally associated with methods
implemented either in the frequency domain, or in the time domain
Sampling can be viewed in a broader sense of projection onto any
subspace or union of subspaces
Can choose the subspaces to yield interesting new possibilities (below
Nyquist sampling of sparse signals, pointwise samples of non
bandlimited signals, perfect compensation of nonlinear effects …)
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Bandlimited Sampling Theorems
Cauchy (1841):

Whittaker (1915) - Shannon (1948):

A. J. Jerri, “The Shannon sampling theorem - its various extensions and applications:
A tutorial review”, Proc. IEEE, pp. 1565-1595, Nov. 1977.
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Limitations of Shannon’s Theorem
Ideal
sampling

Input
bandlimited
Impractical
reconstruction (sinc)
Towards more robust DSPs:
General inputs
Nonideal sampling: general pre-filters, nonlinear distortions
Simple interpolation kernels
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Sampling Process
Linear Distortion
Sampling
functions
Generalized antialiasing filter
Local averaging
Electrical circuit
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Sampling Process
Nonlinear Distortion
Original + Initial guess
Nonlinear
distortion
Linear
distortion
Reconstructed signal
Replace Fourier analysis by functional analysis, Hilbert space
algebra, and convex optimization
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Sampling Process
Noise
The statistical connection:
Employ estimation techniques (different course …)
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Signal Priors
bandlimited
x(t) piece-wise
linear
Different priors lead to different reconstructions
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Signal Priors
Subspace Priors
X( f )
x(t )
Bandlimited
x(t )
Spline spaces
Shift invariant subspace:
Common in communication: pulse amplitude modulation (PAM)
General subspace in a Hilbert space
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Beyond Bandlimited
Two key ideas in bandlimited sampling:
Avoid aliasing
Fourier domain analysis
Misleading concepts!
Suppose that
with
Signal is clearly not bandlimited
Aliasing in frequency and time
Perfect reconstruction possible from samples

Aliasing is not the issue …
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Example: Bandlimited Sampling
Can
be recovered even though
it is not bandlimited?
YES !
1. Compute convolutional inverse of
2. Convolve the samples with
3. Reconstruct with
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Signal Priors
Smoothness Priors
Minimize the worst-case difference:
Infinite dimensional non-convex optimization problem …
Complicated problem but … simple solution

optimal interpolation kernel
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Signal Priors
Stochastic Priors
Original Image
Bicubic Interpolation
Matern Interpolation
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Signal Priors
Sparsity Priors
Wavelet transform of images is commonly sparse
STFT transform of speech signals is commonly sparse
Fourier transform of radio signals is commonly sparse
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Sparse Modelling - Motivation
“Can we not just directly measure the part that will not end up being
thrown away ?”
Original 2500 KB
100%
Donoho
Compressed 148
392
950 KB
15%
6%
38%
OR: It works in digital… Can it work in analog ?
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Compressed Sensing
“Can we not just directly measure the part that will not end up being
thrown away ?”
Donoho
“sensing … as a way of extracting information about an object from a
small number of randomly selected observations”
Candès et. al.
Analog
Audio
Signal
Nyquist rate
Sampling
High-rate
Compressed
Sensing
Compression
(e.g. MP3)
Low-rate
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The Modulated Wideband Converter
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Does The Dream Come True ?
5th order Chebyshev Type-I Filter
Spectrum Analyzer
Sign wavefor generator @ 54 MHz
M=32
Scope
Signal generator
carrier @ 500 MHz
(11 dBm)
Power splitter
(m=2 channels)
Mixer
Lowpass
filter
MATLAB ™
(reconstruction)
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Reconstruction Constraints
Unconstrained Schemes

Sampling
Reconstruction
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Reconstruction Constraints
Predefined Kernel
Predefined

Sampling
Reconstruction
Minimax methods
Consistency requirement
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Reconstruction Constraints
Dense Grid Interpolation
To improve performance: Increase reconstruction rate
Predefined
(e.g. linear interpolation)

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Reconstruction Constraints
Dense Grid Interpolation
Bicubic Interpolation
Second Order
Approximation to Matern
Interpolation with K=2
Optimal Dense Grid Matern
Interpolation with K=2
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Course Outline
(Subject to change without further notice)
Motivating introduction after which you will all want to take this course (1
lesson)
Crash course on linear algebra (basically no prior knowledge is assumed
but strong interest in algebra is highly recommended) (~3 lessons)
Subspace sampling (sampling of nonbandlimited signals, interpolation
methods) (~2 lessons)
Nonlinear sampling (~1 lesson)
Minimax recovery techniques (~1 lesson)
Constrained reconstruction: minimax and consistent methods (~2 lessons)
Sampling sparse signals (1 lesson)
Sampling random signals (1 lesson)
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